How to make material from multiple objects appear as one. Ignore interior faces?
This should probably have been a comment, but I can't create those for now...

If you want to combine objects while eliminating the interior surfaces, the boolean 'union' operator seems the most appropriate to combine your circles. This operator removes internal faces (creating vertices where needed).
It's probably going to be quite fastidious to do, unless you can do it in stages where you combine a few tori, and then duplicate those units (e.g. create units of 4 red circles and one purple).